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40.3k
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5.15k
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100
|
|---|---|---|---|
19,500 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=1+t\cos\alpha}\\{y=t\sin\alpha}\end{array}}\right.$ ($t$ is the parameter, $0\leqslant \alpha\ \ \lt \pi$). Taking the origin $O$ as the pole and the non-negative $x$-axis as the polar axis, the polar equation of curve $C_{2}$ is ${\rho^2}=\frac{{12}}{{3+{{\sin}^2}\theta}}$. <br/>$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of $C_{2}$; <br/>$(2)$ Given $F(1,0)$, the intersection points $A$ and $B$ of curve $C_{1}$ and $C_{2}$ satisfy $|BF|=2|AF|$ (point $A$ is in the first quadrant), find the value of $\cos \alpha$.
|
\frac{2}{3}
| 18.75 |
19,501 |
Nine balls, numbered $1, 2, \cdots, 9$, are placed in a bag. These balls are identical except for their numbers. Person A draws a ball from the bag, which has the number $a$. After placing it back, person B draws another ball from the bag, which has the number $b$. The probability that the inequality $a - 2b + 10 > 0$ holds is
|
$\frac{61}{81}$
| 0 |
19,502 |
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?
|
192
| 3.90625 |
19,503 |
Given that $a$ and $b$ are constants, and $a \neq 0$, $f(x) = ax^2 + bx$, $f(2) = 0$.
$(1)$ If the equation $f(x) - x = 0$ has a unique real root, find the expression for the function $f(x)$;
$(2)$ When $a = 1$, find the maximum and minimum values of the function $f(x)$ in the interval $[-1, 2]$.
|
-1
| 87.5 |
19,504 |
Given the function $f(x)=\cos^2x-\sin^2x+\frac{1}{2}, x \in (0,\pi)$.
$(1)$ Find the interval of monotonic increase for $f(x)$;
$(2)$ Suppose $\triangle ABC$ is an acute triangle, with the side opposite to angle $A$ being $a=\sqrt{19}$, and the side opposite to angle $B$ being $b=5$. If $f(A)=0$, find the area of $\triangle ABC$.
|
\frac{15\sqrt{3}}{4}
| 63.28125 |
19,505 |
Given $x$, $y \in \mathbb{R}^{+}$ and $2x+3y=1$, find the minimum value of $\frac{1}{x}+ \frac{1}{y}$.
|
5+2 \sqrt{6}
| 81.25 |
19,506 |
Given the function $f(x)=2\sin (2x+ \frac {\pi}{4})$, let $f_1(x)$ denote the function after translating and transforming $f(x)$ to the right by $φ$ units and compressing every point's abscissa to half its original length, then determine the minimum value of $φ$ for which $f_1(x)$ is symmetric about the line $x= \frac {\pi}{4}$.
|
\frac{3\pi}{8}
| 60.9375 |
19,507 |
A dice is repeatedly rolled, and the upward-facing number is recorded for each roll. The rolling stops once three different numbers are recorded. If the sequence stops exactly after five rolls, calculate the total number of distinct recording sequences for these five numbers.
|
840
| 16.40625 |
19,508 |
If for any ${x}_{1},{x}_{2}∈[1,\frac{π}{2}]$, $x_{1} \lt x_{2}$, $\frac{{x}_{2}sin{x}_{1}-{x}_{1}sin{x}_{2}}{{x}_{1}-{x}_{2}}＞a$ always holds, then the maximum value of the real number $a$ is ______.
|
-1
| 5.46875 |
19,509 |
A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on Avani's cards?
A 6
B 9
C 10
D 11
E 12
|
12
| 27.34375 |
19,510 |
Given that $a > 0$ and $b > 0$, if the inequality $\frac{3}{a} + \frac{1}{b} \geq \frac{m}{a + 3b}$ always holds true, find the maximum value of $m$.
|
12
| 96.875 |
19,511 |
$\triangle ABC$ has area $240$ . Points $X, Y, Z$ lie on sides $AB$ , $BC$ , and $CA$ , respectively. Given that $\frac{AX}{BX} = 3$ , $\frac{BY}{CY} = 4$ , and $\frac{CZ}{AZ} = 5$ , find the area of $\triangle XYZ$ .
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|
122
| 6.25 |
19,512 |
By joining four identical trapezoids, each with equal non-parallel sides and bases measuring 50 cm and 30 cm, we form a square with an area of 2500 cm² that has a square hole in the middle. What is the area, in cm², of each of the four trapezoids?
|
400
| 22.65625 |
19,513 |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1500$ are not factorial tails?
|
300
| 6.25 |
19,514 |
The set $T = \{1, 2, 3, \ldots, 59, 60\}$ contains the first 60 positive integers. After the multiples of 2, the multiples of 3, and multiples of 5 are removed, how many integers remain in the set $T$?
|
16
| 78.125 |
19,515 |
A certain store sells a type of student backpack. It is known that the cost price of this backpack is $30$ yuan each. Market research shows that the daily sales quantity $y$ (in units) of this backpack is related to the selling price $x$ (in yuan) as follows: $y=-x+60$ ($30\leqslant x\leqslant 60$). Let $w$ represent the daily profit from selling this type of backpack.
(1) Find the functional relationship between $w$ and $x$.
(2) If the pricing department stipulates that the selling price of this backpack should not exceed $48$ yuan, and the store needs to make a daily profit of $200$ yuan from selling this backpack, what should be the selling price?
(3) At what selling price should this backpack be priced to maximize the daily profit? What is the maximum profit?
|
225
| 91.40625 |
19,516 |
A woman buys a property for $15,000, aiming for a $6\%$ return on her investment annually. She sets aside $15\%$ of the rent each month for maintenance, and pays $360 annually in taxes. What must be the monthly rent to meet her financial goals?
A) $110.00$
B) $123.53$
C) $130.45$
D) $142.86$
E) $150.00$
|
123.53
| 71.09375 |
19,517 |
Evaluate $\sqrt[3]{1+27} + \sqrt[3]{1+\sqrt[3]{27}}$.
|
\sqrt[3]{28} + \sqrt[3]{4}
| 72.65625 |
19,518 |
For how many values of \( k \) is \( 18^{18} \) the least common multiple of the positive integers \( 9^9 \), \( 12^{12} \), and \( k \)?
|
19
| 24.21875 |
19,519 |
A cowboy is 5 miles north of a stream which flows due west. He is also 10 miles east and 6 miles south of his cabin. He wishes to water his horse at the stream and then return home. Determine the shortest distance he can travel to accomplish this.
A) $5 + \sqrt{256}$ miles
B) $5 + \sqrt{356}$ miles
C) $11 + \sqrt{356}$ miles
D) $5 + \sqrt{116}$ miles
|
5 + \sqrt{356}
| 10.15625 |
19,520 |
Compute the values of $\binom{600}{600}$, $\binom{600}{0}$, and $\binom{600}{1}$.
|
600
| 21.09375 |
19,521 |
How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?
A sequence of three numbers \( a, b, c \) is said to form an arithmetic progression if \( a + c = 2b \).
A correct numerical answer without justification will earn 4 points. For full points, a detailed reasoning is expected.
|
45
| 99.21875 |
19,522 |
As we enter the autumn and winter seasons, the air becomes dry. A certain appliance store is preparing to purchase a batch of humidifiers. The cost price of each unit is $80$ yuan. After market research, the selling price is set at $100$ yuan per unit. The store can sell $500$ units per day. For every $1$ yuan increase in price, the daily sales volume will decrease by $10$ units. Let $x$ represent the increase in price per unit.
$(1)$ If the daily sales volume is denoted by $y$ units, write down the relationship between $y$ and $x$ directly.
$(2)$ Express the profit $W$ in yuan obtained by the store from selling each humidifier per day using an algebraic expression involving $x$. Calculate the selling price per unit that maximizes profit. What is the maximum profit obtained?
|
12250
| 80.46875 |
19,523 |
In a pentagon ABCDE, there is a vertical line of symmetry. Vertex E is moved to \(E(5,0)\), while \(A(0,0)\), \(B(0,5)\), and \(D(5,5)\). What is the \(y\)-coordinate of vertex C such that the area of pentagon ABCDE becomes 65 square units?
|
21
| 47.65625 |
19,524 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$.
|
\frac{1}{72}
| 11.71875 |
19,525 |
Given the set $A=\{x\in \mathbb{R} | ax^2-3x+2=0, a\in \mathbb{R}\}$.
1. If $A$ is an empty set, find the range of values for $a$.
2. If $A$ contains only one element, find the value of $a$ and write down this element.
|
\frac{4}{3}
| 60.15625 |
19,526 |
A circle is tangent to the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $AB$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is nine times smaller than the other.
|
12
| 14.84375 |
19,527 |
Billy's age is three times Brenda's age and twice Joe's age. The sum of their ages is 72. How old is Billy?
|
\frac{432}{11}
| 2.34375 |
19,528 |
Given the sequence \(\{a_n\}\) with the sum of its first \(n\) terms denoted by \(S_n\), let \(T_n = \frac{S_1 + S_2 + \cdots + S_n}{n}\). \(T_n\) is called the "mean" of the sequence \(a_1, a_2, \cdots, a_n\). It is known that the "mean" of the sequence \(a_1, a_2, \cdots, a_{1005}\) is 2012. Determine the "mean" of the sequence \(-1, a_1, a_2, \cdots, a_{1005}\).
|
2009
| 25 |
19,529 |
Given the geometric sequence $\{a_n\}$, $a_5a_7=2$, $a_2+a_{10}=3$, determine the value of $\frac{a_{12}}{a_4}$.
|
\frac {1}{2}
| 17.1875 |
19,530 |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $c$ denote the number of positive integers $n \leq 1000$ with $S(n)$ odd, and let $d$ denote the number of positive integers $n \leq 1000$ with $S(n)$ even. Find $|c-d|.$
|
33
| 0 |
19,531 |
Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus.
|
\frac{90\sqrt{261}}{261}
| 0.78125 |
19,532 |
Given: In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $\frac {\cos A-2\cos C}{\cos B}= \frac {2c-a}{b}$.
$(1)$ Find the value of $\frac {\sin C}{\sin A}$;
$(2)$ If $\cos B= \frac {1}{4}$ and $b=2$, find the area $S$ of $\triangle ABC$.
|
\frac { \sqrt {15}}{4}
| 0 |
19,533 |
Each outcome on the spinner below has equal probability. If you spin the spinner four times and form a four-digit number from the four outcomes, such that the first outcome is the thousand digit, the second outcome is the hundreds digit, the third outcome is the tens digit, and the fourth outcome is the units digit, what is the probability that you will end up with a four-digit number that is divisible by 5? Use the spinner with digits 0, 1, and 5.
|
\frac{2}{3}
| 95.3125 |
19,534 |
Given a list of the first 12 positive integers such that for each $2\le i\le 12$, either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list, calculate the number of such lists.
|
2048
| 15.625 |
19,535 |
Given the function $f(x) = 4\sin^2 x + \sin\left(2x + \frac{\pi}{6}\right) - 2$,
$(1)$ Determine the interval over which $f(x)$ is strictly decreasing;
$(2)$ Find the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$ and determine the value(s) of $x$ at which the maximum value occurs.
|
\frac{5\pi}{12}
| 12.5 |
19,536 |
In triangle $ABC$, angle $A$ is $90^\circ$, $BC = 10$ and $\tan C = 3\cos B$. What is $AB$?
|
\frac{20\sqrt{2}}{3}
| 57.03125 |
19,537 |
In a store, there are 50 light bulbs in stock, 60% of which are produced by Factory A and 40% by Factory B. The first-class rate of the light bulbs produced by Factory A is 90%, and the first-class rate of the light bulbs produced by Factory B is 80%.
(1) If one light bulb is randomly selected from these 50 light bulbs (each light bulb has an equal chance of being selected), what is the probability that it is a first-class product produced by Factory A?
(2) If two light bulbs are randomly selected from these 50 light bulbs (each light bulb has an equal chance of being selected), and the number of first-class products produced by Factory A among these two light bulbs is denoted as $\xi$, find the value of $E(\xi)$.
|
1.08
| 21.875 |
19,538 |
The minimum positive period of the function $y=\sin x \cdot |\cos x|$ is __________.
|
2\pi
| 17.96875 |
19,539 |
Given the curve $\frac{y^{2}}{b} - \frac{x^{2}}{a} = 1 (a \cdot b \neq 0, a \neq b)$ and the line $x + y - 2 = 0$, the points $P$ and $Q$ intersect at the curve and line, and $\overrightarrow{OP} \cdot \overrightarrow{OQ} = 0 (O$ is the origin$), then the value of $\frac{1}{b} - \frac{1}{a}$ is $\_\_\_\_\_\_\_\_\_$.
|
\frac{1}{2}
| 75 |
19,540 |
Let
\[x^6 - x^3 - x^2 - x - 1 = q_1(x) q_2(x) \dotsm q_m(x),\]
where each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(2) + q_2(2) + \dots + q_m(2).$
|
14
| 5.46875 |
19,541 |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$ , $y$ , and $z$ . Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$ , $17$ , $21$ , and $23$ . Compute $x^2+y^2+z^2$ .
|
371
| 50 |
19,542 |
What is the greatest common divisor of $8!$ and $10!$?
|
40320
| 100 |
19,543 |
Compute
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples \((a,b,c)\) of positive integers such that \(1 \le a < b < c\).)
|
\frac{1}{21216}
| 48.4375 |
19,544 |
Given \( \cos \left( \frac {\pi}{2}+\alpha \right)=3\sin \left(\alpha+ \frac {7\pi}{6}\right) \), find the value of \( \tan \left( \frac {\pi}{12}+\alpha \right) = \) ______.
|
2\sqrt {3} - 4
| 0 |
19,545 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases}x=3\cos \alpha \\ y=\sin \alpha\end{cases}$ (where $\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\rho\sin \left( \theta -\frac{\pi }{4} \right)=\sqrt{2}$.
(1) Find the standard equation of $C$ and the inclination angle of line $l$;
(2) Let point $P(0,2)$, line $l$ intersects curve $C$ at points $A$ and $B$, find $|PA|+|PB|$.
|
\frac{18 \sqrt{2}}{5}
| 28.90625 |
19,546 |
Suppose a point $P$ has coordinates $(m, n)$, where $m$ and $n$ are the points obtained by rolling a dice twice consecutively. The probability that point $P$ lies outside the circle $x^{2}+y^{2}=16$ is _______.
|
\frac {7}{9}
| 3.90625 |
19,547 |
Given Ben's test scores $95, 85, 75, 65,$ and $90$, and his goal to increase his average by at least $5$ points and score higher than his lowest score of $65$ with his next test, calculate the minimum test score he would need to achieve both goals.
|
112
| 73.4375 |
19,548 |
Given a tesseract (4-dimensional hypercube), calculate the sum of the number of edges, vertices, and faces.
|
72
| 95.3125 |
19,549 |
The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits
|
99972
| 100 |
19,550 |
Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$ . Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$ . If $\angle MPN = 40^\circ$ , find the degree measure of $\angle BPC$ .
*Ray Li.*
|
80
| 35.9375 |
19,551 |
Given that points $E$ and $F$ are on the same side of diameter $\overline{GH}$ in circle $P$, $\angle GPE = 60^\circ$, and $\angle FPH = 90^\circ$, find the ratio of the area of the smaller sector $PEF$ to the area of the circle.
|
\frac{1}{12}
| 27.34375 |
19,552 |
Given the function $f(x)=4\cos x\cos \left(x- \frac {\pi}{3}\right)-2$.
$(I)$ Find the smallest positive period of the function $f(x)$.
$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $\left[- \frac {\pi}{6}, \frac {\pi}{4}\right]$.
|
-2
| 64.84375 |
19,553 |
A construction company purchased a piece of land for 80 million yuan. They plan to build a building with at least 12 floors on this land, with each floor having an area of 4000 square meters. Based on preliminary estimates, if the building is constructed with x floors (where x is greater than or equal to 12 and x is a natural number), then the average construction cost per square meter is given by s = 3000 + 50x (in yuan). In order to minimize the average comprehensive cost per square meter W (in yuan), which includes both the average construction cost and the average land purchase cost per square meter, the building should have how many floors? What is the minimum value of the average comprehensive cost per square meter? Note: The average comprehensive cost per square meter equals the average construction cost per square meter plus the average land purchase cost per square meter, where the average land purchase cost per square meter is calculated as the total land purchase cost divided by the total construction area (pay attention to unit consistency).
|
5000
| 73.4375 |
19,554 |
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, $c$, respectively, and $C=\frac{π}{3}$, $c=2$. Then find the maximum value of $\overrightarrow{AC}•\overrightarrow{AB}$.
|
\frac{4\sqrt{3}}{3} + 2
| 0 |
19,555 |
In the plane rectangular coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system with the same unit of length. The parametric equation of line $l$ is $\begin{cases}x=2+\frac{\sqrt{2}}{2}t\\y=1+\frac{\sqrt{2}}{2}t\end{cases}$, and the polar coordinate equation of circle $C$ is $\rho=4\sqrt{2}\sin\left(\theta+\frac{\pi}{4}\right)$.
(1) Find the ordinary equation of line $l$ and the rectangular coordinate equation of circle $C$.
(2) Suppose curve $C$ intersects with line $l$ at points $A$ and $B$. If the rectangular coordinate of point $P$ is $(2,1)$, find the value of $||PA|-|PB||$.
|
\sqrt{2}
| 65.625 |
19,556 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. If $\cos 2B + \cos B + \cos (A-C) = 1$ and $b = \sqrt{7}$, find the minimum value of $a^2 + c^2$.
|
14
| 41.40625 |
19,557 |
ABCD is a square. BDEF is a rhombus with A, E, and F collinear. Find ∠ADE.
|
15
| 0.78125 |
19,558 |
Last year, a bicycle cost $200, a cycling helmet $50, and a water bottle $15. This year the cost of each has increased by 6% for the bicycle, 12% for the helmet, and 8% for the water bottle respectively. Find the percentage increase in the combined cost of the bicycle, helmet, and water bottle.
A) $6.5\%$
B) $7.25\%$
C) $7.5\%$
D) $8\%$
|
7.25\%
| 59.375 |
19,559 |
John has two identical cups. Initially, he puts 6 ounces of tea into the first cup and 6 ounces of milk into the second cup. He then pours one-third of the tea from the first cup into the second cup and mixes thoroughly. After stirring, John then pours half of the mixture from the second cup back into the first cup. Finally, he pours one-quarter of the mixture from the first cup back into the second cup. What fraction of the liquid in the first cup is now milk?
|
\frac{3}{8}
| 36.71875 |
19,560 |
Reading material: After studying square roots, Xiaoming found that some expressions containing square roots can be written as the square of another expression, such as: $3+2\sqrt{2}=(1+\sqrt{2})^{2}$. With his good thinking skills, Xiaoming conducted the following exploration:<br/>Let: $a+b\sqrt{2}=(m+n\sqrt{2})^2$ (where $a$, $b$, $m$, $n$ are all integers), then we have $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$.<br/>$\therefore a=m^{2}+2n^{2}$, $b=2mn$. In this way, Xiaoming found a method to convert some expressions of $a+b\sqrt{2}$ into square forms. Please follow Xiaoming's method to explore and solve the following problems:<br/>$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=(m+n\sqrt{3})^2$, express $a$, $b$ in terms of $m$, $n$, and get $a=$______, $b=$______;<br/>$(2)$ Using the conclusion obtained, find a set of positive integers $a$, $b$, $m$, $n$, fill in the blanks: ______$+\_\_\_\_\_\_=( \_\_\_\_\_\_+\_\_\_\_\_\_\sqrt{3})^{2}$;<br/>$(3)$ If $a+4\sqrt{3}=(m+n\sqrt{3})^2$, and $a$, $b$, $m$, $n$ are all positive integers, find the value of $a$.
|
13
| 18.75 |
19,561 |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively, and it is given that $a < b < c$ and $$\frac{a}{\sin A} = \frac{2b}{\sqrt{3}}$$.
(1) Find the size of angle $B$;
(2) If $a=2$ and $c=3$, find the length of side $b$ and the area of $\triangle ABC$.
|
\frac{3\sqrt{3}}{2}
| 99.21875 |
19,562 |
Given the functions $f(x)=x^{2}-2x+m\ln x(m∈R)$ and $g(x)=(x- \frac {3}{4})e^{x}$.
(1) If $m=-1$, find the value of the real number $a$ such that the minimum value of the function $φ(x)=f(x)-\[x^{2}-(2+ \frac {1}{a})x\](0 < x\leqslant e)$ is $2$;
(2) If $f(x)$ has two extreme points $x_{1}$, $x_{2}(x_{1} < x_{2})$, find the minimum value of $g(x_{1}-x_{2})$.
|
-e^{- \frac {1}{4}}
| 27.34375 |
19,563 |
Calculate:<br/>$(1)(\sqrt{3})^2+|1-\sqrt{3}|+\sqrt[3]{-27}$;<br/>$(2)(\sqrt{12}-\sqrt{\frac{1}{3}})×\sqrt{6}$.
|
5\sqrt{2}
| 84.375 |
19,564 |
What are the last two digits in the sum of the factorials of the first 15 positive integers?
|
13
| 35.9375 |
19,565 |
Given the function $f(x)=2\sin x\cos x+1-2\sin^2x$.
(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$.
|
-\frac{\sqrt{3}+1}{2}
| 47.65625 |
19,566 |
Let \(ABCD\) be a quadrilateral circumscribed about a circle with center \(O\). Let \(O_1, O_2, O_3,\) and \(O_4\) denote the circumcenters of \(\triangle AOB, \triangle BOC, \triangle COD,\) and \(\triangle DOA\). If \(\angle A = 120^\circ\), \(\angle B = 80^\circ\), and \(\angle C = 45^\circ\), what is the acute angle formed by the two lines passing through \(O_1 O_3\) and \(O_2 O_4\)?
|
45
| 12.5 |
19,567 |
From a set consisting of three blue cards labeled $X$, $Y$, $Z$ and three orange cards labeled $X$, $Y$, $Z$, two cards are randomly drawn. A winning pair is defined as either two cards with the same label or two cards of the same color. What is the probability of drawing a winning pair?
- **A)** $\frac{1}{3}$
- **B)** $\frac{1}{2}$
- **C)** $\frac{3}{5}$
- **D)** $\frac{2}{3}$
- **E)** $\frac{4}{5}$
|
\frac{3}{5}
| 64.0625 |
19,568 |
What is the ratio of the area of the shaded square to the area of the large square given that the large square is divided into a grid of $25$ smaller squares each of side $1$ unit? The shaded square consists of a pentagon formed by connecting the centers of adjacent half-squares on the grid. The shaded region is not provided in a diagram but is described as occupying $5$ half-squares in the center of the large square.
|
\frac{1}{10}
| 83.59375 |
19,569 |
If the vertices of a smaller square are midpoints of the sides of a larger square, and the larger square has an area of 144, what is the area of the smaller square?
|
72
| 85.9375 |
19,570 |
Two circles of radii 4 and 5 are externally tangent to each other and are circumscribed by a third circle. Find the area of the shaded region created in this way. Express your answer in terms of $\pi$.
|
40\pi
| 14.0625 |
19,571 |
Given real numbers $x$ and $y$ satisfying $\sqrt{x-3}+y^{2}-4y+4=0$, find the value of the algebraic expression $\frac{{x}^{2}-{y}^{2}}{xy}•\frac{1}{{x}^{2}-2xy+{y}^{2}}÷\frac{x}{{x}^{2}y-x{y}^{2}}-1$.
|
\frac{2}{3}
| 75.78125 |
19,572 |
How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8?
|
14
| 3.125 |
19,573 |
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, and ${S}_{n}=2{a}_{n}-{2}^{n+1}$. The sequence $\{b_{n}\}$ satisfies ${b}_{n}=log_{2}\frac{{a}_{n}}{n+1}$, where $n\in N^{*}$. Find the maximum real number $m$ such that the inequality $(1+\frac{1}{{b}_{2}})•(1+\frac{1}{{b}_{4}})•⋯•(1+\frac{1}{{b}_{2n}})≥m•\sqrt{{b}_{2n+2}}$ holds for all positive integers $n$.
|
\frac{3}{4}
| 19.53125 |
19,574 |
Calculate $\left(\sqrt{(\sqrt{5})^4}\right)^6$.
|
15625
| 96.09375 |
19,575 |
Given $\sin (30^{\circ}+\alpha)= \frac {3}{5}$, and $60^{\circ} < \alpha < 150^{\circ}$, solve for the value of $\cos \alpha$.
|
\frac{3-4\sqrt{3}}{10}
| 43.75 |
19,576 |
The function $g(x)$ satisfies the equation
\[xg(y) = 2yg(x)\] for all real numbers $x$ and $y$. If $g(10) = 30$, find $g(2)$.
|
12
| 32.03125 |
19,577 |
Given the data set $(4.7)$, $(4.8)$, $(5.1)$, $(5.4)$, $(5.5)$, calculate the variance of the data set.
|
0.1
| 95.3125 |
19,578 |
Find the square root of $\dfrac{9!}{126}$.
|
12\sqrt{10}
| 0.78125 |
19,579 |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$ ?
|
16
| 93.75 |
19,580 |
There are 20 cards with numbers $1, 2, \cdots, 19, 20$ on them. These cards are placed in a box, and 4 people each draw one card. The two people who draw the smaller numbers will be in one group, and the two who draw the larger numbers will be in another group. If two of the people draw the numbers 5 and 14 respectively, what is the probability that these two people will be in the same group?
|
$\frac{7}{51}$
| 0 |
19,581 |
Given that $\sin{\alpha} = -\frac{3}{5}$, $\sin{\beta} = \frac{12}{13}$, and $\alpha \in (\pi, \frac{3\pi}{2})$, $\beta \in (\frac{\pi}{2}, \pi)$, find the values of $\sin({\alpha - \beta})$, $\cos{2\alpha}$, and $\tan{\frac{\beta}{2}}$.
|
\frac{3}{2}
| 62.5 |
19,582 |
Given \(\sin^{2}(18^{\circ}) + \cos^{2}(63^{\circ}) + \sqrt{2} \sin(18^{\circ}) \cdot \cos(63^{\circ}) =\)
|
\frac{1}{2}
| 11.71875 |
19,583 |
Given vectors $\overrightarrow{a}=(2\cos \alpha,\sin ^{2}\alpha)$, $\overrightarrow{b}=(2\sin \alpha,t)$, where $\alpha\in(0, \frac {\pi}{2})$, and $t$ is a real number.
$(1)$ If $\overrightarrow{a}- \overrightarrow{b}=( \frac {2}{5},0)$, find the value of $t$;
$(2)$ If $t=1$, and $\overrightarrow{a}\cdot \overrightarrow{b}=1$, find the value of $\tan (2\alpha+ \frac {\pi}{4})$.
|
\frac {23}{7}
| 6.25 |
19,584 |
In a shooting competition, there are 8 clay targets arranged in 3 columns as shown in the diagram. A sharpshooter follows these rules to shoot down all the targets:
1. First, select a column from which one target will be shot.
2. Then, target the lowest remaining target in the selected column.
How many different sequences are there to shoot down all 8 targets?
|
560
| 75.78125 |
19,585 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\left(a+b+c\right)\left(a+b-c\right)=ab$.
$(1)$ Find angle $C$;
$(2)$ If the angle bisector of angle $C$ intersects $AB$ at point $D$ and $CD=2$, find the minimum value of $2a+b$.
|
6+4\sqrt{2}
| 35.15625 |
19,586 |
Find the simplest method to solve the system of equations using substitution $$\begin{cases} x=2y\textcircled{1} \\ 2x-y=5\textcircled{2} \end{cases}$$
|
y = \frac{5}{3}
| 0 |
19,587 |
**How many positive factors does 72 have, and what is their sum?**
|
195
| 55.46875 |
19,588 |
Given that the odometer initially displays $12321$ miles, and after $4$ hours, the next higher palindrome is displayed, calculate the average speed in miles per hour during this $4$-hour period.
|
25
| 53.125 |
19,589 |
Consider two fictional states: Alpha and Beta. Alpha issues license plates with a format of two letters followed by four numbers, and then ending with one letter (LLNNNNL). Beta issues plates with three letters followed by three numbers and lastly one letter (LLLNNNL). Assume all 10 digits and 26 letters are equally likely to appear in the respective slots. How many more license plates can state Alpha issue than state Beta?
|
281216000
| 19.53125 |
19,590 |
The function $f$ satisfies
\[
f(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3
\]
for all real numbers $x, y$. Determine the value of $f(10)$.
|
-37
| 22.65625 |
19,591 |
Calculate the perimeter of a triangle whose vertices are located at points $A(2,3)$, $B(2,10)$, and $C(8,6)$ on a Cartesian coordinate plane.
|
7 + 2\sqrt{13} + 3\sqrt{5}
| 89.84375 |
19,592 |
$\triangle KWU$ is an equilateral triangle with side length $12$ . Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$ . If $\overline{KP} = 13$ , find the length of the altitude from $P$ onto $\overline{WU}$ .
*Proposed by Bradley Guo*
|
\frac{25\sqrt{3}}{24}
| 0 |
19,593 |
Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$?
|
4.5
| 0.78125 |
19,594 |
Find the number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889.)
|
75582
| 14.0625 |
19,595 |
Given the function $f(x)=\sin ( \frac {7π}{6}-2x)-2\sin ^{2}x+1(x∈R)$,
(1) Find the period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f(x)$ passes through points $(A, \frac {1}{2}),b,a,c$ forming an arithmetic sequence, and $\overrightarrow{AB}\cdot \overrightarrow{AC}=9$, find the value of $a$.
|
3 \sqrt {2}
| 0 |
19,596 |
In right triangle $XYZ$, we have $\angle X = \angle Z$ and $XZ = 8\sqrt{2}$. What is the area of $\triangle XYZ$?
|
32
| 84.375 |
19,597 |
The product of two positive integers plus their sum equals 119. The integers are relatively prime, and each is less than 25. What is the sum of the two integers?
|
20
| 24.21875 |
19,598 |
In a recent survey conducted by Mary, she found that $72.4\%$ of participants believed that rats are typically blind. Among those who held this belief, $38.5\%$ mistakenly thought that all rats are albino, which is not generally true. Mary noted that 25 people had this specific misconception. Determine how many total people Mary surveyed.
|
90
| 64.84375 |
19,599 |
Given the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with $a > b > 0$ and eccentricity $e = \frac{\sqrt{5}-1}{2}$, find the product of the slopes of lines $PA$ and $PB$ for a point $P$ on the ellipse that is not the left or right vertex.
|
\frac{1-\sqrt{5}}{2}
| 52.34375 |
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